#### Volume 21, issue 3 (2017)

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Homological stability for spaces of embedded surfaces

### Federico Cantero Morán and Oscar Randal-Williams

Geometry & Topology 21 (2017) 1387–1467
##### Abstract

We study the space of oriented genus-$g$ subsurfaces of a fixed manifold $M$ and, in particular, its homological properties. We construct a “scanning map” which compares this space to the space of sections of a certain fibre bundle over $M$ associated to its tangent bundle, and show that this map induces an isomorphism on homology in a range of degrees.

Our results are analogous to McDuff’s theorem on configuration spaces, extended from $0$–dimensional submanifolds to $2$–dimensional submanifolds.

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##### Keywords
submanifolds, characteristic classes, homology stability, embedding spaces, mapping class groups, scanning
##### Mathematical Subject Classification 2010
Primary: 55R40, 57R20, 57R40, 57R50, 57S05